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Chi-Square Test of Homogeneity

Slide Duration:

Section 1: Introduction
Descriptive Statistics vs. Inferential Statistics

25m 31s

Intro
0:00
0:10
0:11
Statistics
0:35
Statistics
0:36
Let's Think About High School Science
1:12
Measurement and Find Patterns (Mathematical Formula)
1:13
Statistics = Math of Distributions
4:58
Distributions
4:59
Problematic… but also GREAT
5:58
Statistics
7:33
How is It Different from Other Specializations in Mathematics?
7:34
Statistics is Fundamental in Natural and Social Sciences
7:53
Two Skills of Statistics
8:20
Description (Exploration)
8:21
Inference
9:13
Descriptive Statistics vs. Inferential Statistics: Apply to Distributions
9:58
Descriptive Statistics
9:59
Inferential Statistics
11:05
Populations vs. Samples
12:19
Populations vs. Samples: Is it the Truth?
12:20
Populations vs. Samples: Pros & Cons
13:36
Populations vs. Samples: Descriptive Values
16:12
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:10
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:11
Example 1: Descriptive Statistics vs. Inferential Statistics
19:09
Example 2: Descriptive Statistics vs. Inferential Statistics
20:47
Example 3: Sample, Parameter, Population, and Statistic
21:40
Example 4: Sample, Parameter, Population, and Statistic
23:28
Section 2: About Samples: Cases, Variables, Measurements

32m 14s

Intro
0:00
Data
0:09
Data, Cases, Variables, and Values
0:10
Rows, Columns, and Cells
2:03
Example: Aircrafts
3:52
How Do We Get Data?
5:38
Research: Question and Hypothesis
5:39
Research Design
7:11
Measurement
7:29
Research Analysis
8:33
Research Conclusion
9:30
Types of Variables
10:03
Discrete Variables
10:04
Continuous Variables
12:07
Types of Measurements
14:17
Types of Measurements
14:18
Types of Measurements (Scales)
17:22
Nominal
17:23
Ordinal
19:11
Interval
21:33
Ratio
24:24
Example 1: Cases, Variables, Measurements
25:20
Example 2: Which Scale of Measurement is Used?
26:55
Example 3: What Kind of a Scale of Measurement is This?
27:26
Example 4: Discrete vs. Continuous Variables.
30:31
Section 3: Visualizing Distributions
Introduction to Excel

8m 9s

Intro
0:00
Before Visualizing Distribution
0:10
Excel
0:11
Excel: Organization
0:45
Workbook
0:46
Column x Rows
1:50
Tools: Menu Bar, Standard Toolbar, and Formula Bar
3:00
Excel + Data
6:07
Exce and Data
6:08
Frequency Distributions in Excel

39m 10s

Intro
0:00
0:08
Data in Excel and Frequency Distributions
0:09
Raw Data to Frequency Tables
0:42
Raw Data to Frequency Tables
0:43
Frequency Tables: Using Formulas and Pivot Tables
1:28
Example 1: Number of Births
7:17
Example 2: Age Distribution
20:41
Example 3: Height Distribution
27:45
Example 4: Height Distribution of Males
32:19
Frequency Distributions and Features

25m 29s

Intro
0:00
0:10
Data in Excel, Frequency Distributions, and Features of Frequency Distributions
0:11
Example #1
1:35
Uniform
1:36
Example #2
2:58
Unimodal, Skewed Right, and Asymmetric
2:59
Example #3
6:29
Bimodal
6:30
Example #4a
8:29
Symmetric, Unimodal, and Normal
8:30
Point of Inflection and Standard Deviation
11:13
Example #4b
12:43
Normal Distribution
12:44
Summary
13:56
Uniform, Skewed, Bimodal, and Normal
13:57
17:34
Sketch Problem 2: Life Expectancy
20:01
Sketch Problem 3: Telephone Numbers
22:01
Sketch Problem 4: Length of Time Used to Complete a Final Exam
23:43
Dotplots and Histograms in Excel

42m 42s

Intro
0:00
0:06
0:07
Previously
1:02
Data, Frequency Table, and visualization
1:03
Dotplots
1:22
Dotplots Excel Example
1:23
Dotplots: Pros and Cons
7:22
Pros and Cons of Dotplots
7:23
Dotplots Excel Example Cont.
9:07
Histograms
12:47
Histograms Overview
12:48
Example of Histograms
15:29
Histograms: Pros and Cons
31:39
Pros
31:40
Cons
32:31
Frequency vs. Relative Frequency
32:53
Frequency
32:54
Relative Frequency
33:36
Example 1: Dotplots vs. Histograms
34:36
Example 2: Age of Pennies Dotplot
36:21
Example 3: Histogram of Mammal Speeds
38:27
Example 4: Histogram of Life Expectancy
40:30
Stemplots

12m 23s

Intro
0:00
0:05
0:06
What Sets Stemplots Apart?
0:46
Data Sets, Dotplots, Histograms, and Stemplots
0:47
Example 1: What Do Stemplots Look Like?
1:58
Example 2: Back-to-Back Stemplots
5:00
7:46
Example 4: Quiz Grade & Afterschool Tutoring Stemplot
9:56
Bar Graphs

22m 49s

Intro
0:00
0:05
0:08
Review of Frequency Distributions
0:44
Y-axis and X-axis
0:45
Types of Frequency Visualizations Covered so Far
2:16
Introduction to Bar Graphs
4:07
Example 1: Bar Graph
5:32
Example 1: Bar Graph
5:33
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:07
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:08
Example 2: Create a Frequency Visualization for Gender
14:02
Example 3: Cases, Variables, and Frequency Visualization
16:34
Example 4: What Kind of Graphs are Shown Below?
19:29
Section 4: Summarizing Distributions
Central Tendency: Mean, Median, Mode

38m 50s

Intro
0:00
0:07
0:08
Central Tendency 1
0:56
Way to Summarize a Distribution of Scores
0:57
Mode
1:32
Median
2:02
Mean
2:36
Central Tendency 2
3:47
Mode
3:48
Median
4:20
Mean
5:25
Summation Symbol
6:11
Summation Symbol
6:12
Population vs. Sample
10:46
Population vs. Sample
10:47
Excel Examples
15:08
Finding Mode, Median, and Mean in Excel
15:09
Median vs. Mean
21:45
Effect of Outliers
21:46
Relationship Between Parameter and Statistic
22:44
Type of Measurements
24:00
Which Distributions to Use With
24:55
Example 1: Mean
25:30
Example 2: Using Summation Symbol
29:50
Example 3: Average Calorie Count
32:50
Example 4: Creating an Example Set
35:46
Variability

42m 40s

Intro
0:00
0:05
0:06
0:45
0:46
5:45
5:46
Range, Quartiles and Interquartile Range
6:37
Range
6:38
Interquartile Range
8:42
Interquartile Range Example
10:58
Interquartile Range Example
10:59
Variance and Standard Deviation
12:27
Deviations
12:28
Sum of Squares
14:35
Variance
16:55
Standard Deviation
17:44
Sum of Squares (SS)
18:34
Sum of Squares (SS)
18:35
Population vs. Sample SD
22:00
Population vs. Sample SD
22:01
Population vs. Sample
23:20
Mean
23:21
SD
23:51
Example 1: Find the Mean and Standard Deviation of the Variable Friends in the Excel File
27:21
Example 2: Find the Mean and Standard Deviation of the Tagged Photos in the Excel File
35:25
Example 3: Sum of Squares
38:58
Example 4: Standard Deviation
41:48
Five Number Summary & Boxplots

57m 15s

Intro
0:00
0:06
0:07
Summarizing Distributions
0:37
0:38
5 Number Summary
1:14
Boxplot: Visualizing 5 Number Summary
3:37
Boxplot: Visualizing 5 Number Summary
3:38
Boxplots on Excel
9:01
Using 'Stocks' and Using Stacked Columns
9:02
Boxplots on Excel Example
10:14
When are Boxplots Useful?
32:14
Pros
32:15
Cons
32:59
How to Determine Outlier Status
33:24
Rule of Thumb: Upper Limit
33:25
Rule of Thumb: Lower Limit
34:16
Signal Outliers in an Excel Data File Using Conditional Formatting
34:52
Modified Boxplot
48:38
Modified Boxplot
48:39
Example 1: Percentage Values & Lower and Upper Whisker
49:10
Example 2: Boxplot
50:10
Example 3: Estimating IQR From Boxplot
53:46
Example 4: Boxplot and Missing Whisker
54:35
Shape: Calculating Skewness & Kurtosis

41m 51s

Intro
0:00
0:16
0:17
Skewness Concept
1:09
Skewness Concept
1:10
Calculating Skewness
3:26
Calculating Skewness
3:27
Interpreting Skewness
7:36
Interpreting Skewness
7:37
Excel Example
8:49
Kurtosis Concept
20:29
Kurtosis Concept
20:30
Calculating Kurtosis
24:17
Calculating Kurtosis
24:18
Interpreting Kurtosis
29:01
Leptokurtic
29:35
Mesokurtic
30:10
Platykurtic
31:06
Excel Example
32:04
Example 1: Shape of Distribution
38:28
Example 2: Shape of Distribution
39:29
Example 3: Shape of Distribution
40:14
Example 4: Kurtosis
41:10
Normal Distribution

34m 33s

Intro
0:00
0:13
0:14
What is a Normal Distribution
0:44
The Normal Distribution As a Theoretical Model
0:45
Possible Range of Probabilities
3:05
Possible Range of Probabilities
3:06
What is a Normal Distribution
5:07
Can Be Described By
5:08
Properties
5:49
'Same' Shape: Illusion of Different Shape!
7:35
'Same' Shape: Illusion of Different Shape!
7:36
Types of Problems
13:45
Example: Distribution of SAT Scores
13:46
Shape Analogy
19:48
Shape Analogy
19:49
Example 1: The Standard Normal Distribution and Z-Scores
22:34
Example 2: The Standard Normal Distribution and Z-Scores
25:54
Example 3: Sketching and Normal Distribution
28:55
Example 4: Sketching and Normal Distribution
32:32
Standard Normal Distributions & Z-Scores

41m 44s

Intro
0:00
0:06
0:07
A Family of Distributions
0:28
Infinite Set of Distributions
0:29
Transforming Normal Distributions to 'Standard' Normal Distribution
1:04
Normal Distribution vs. Standard Normal Distribution
2:58
Normal Distribution vs. Standard Normal Distribution
2:59
Z-Score, Raw Score, Mean, & SD
4:08
Z-Score, Raw Score, Mean, & SD
4:09
Weird Z-Scores
9:40
Weird Z-Scores
9:41
Excel
16:45
For Normal Distributions
16:46
For Standard Normal Distributions
19:11
Excel Example
20:24
Types of Problems
25:18
Percentage Problem: P(x)
25:19
Raw Score and Z-Score Problems
26:28
Standard Deviation Problems
27:01
Shape Analogy
27:44
Shape Analogy
27:45
Example 1: Deaths Due to Heart Disease vs. Deaths Due to Cancer
28:24
Example 2: Heights of Male College Students
33:15
Example 3: Mean and Standard Deviation
37:14
Example 4: Finding Percentage of Values in a Standard Normal Distribution
37:49
Normal Distribution: PDF vs. CDF

55m 44s

Intro
0:00
0:15
0:16
Frequency vs. Cumulative Frequency
0:56
Frequency vs. Cumulative Frequency
0:57
Frequency vs. Cumulative Frequency
4:32
Frequency vs. Cumulative Frequency Cont.
4:33
Calculus in Brief
6:21
Derivative-Integral Continuum
6:22
PDF
10:08
PDF for Standard Normal Distribution
10:09
PDF for Normal Distribution
14:32
Integral of PDF = CDF
21:27
Integral of PDF = CDF
21:28
Example 1: Cumulative Frequency Graph
23:31
Example 2: Mean, Standard Deviation, and Probability
24:43
Example 3: Mean and Standard Deviation
35:50
Example 4: Age of Cars
49:32
Section 5: Linear Regression
Scatterplots

47m 19s

Intro
0:00
0:04
0:05
Previous Visualizations
0:30
Frequency Distributions
0:31
Compare & Contrast
2:26
Frequency Distributions Vs. Scatterplots
2:27
Summary Values
4:53
Shape
4:54
Center & Trend
6:41
8:22
Univariate & Bivariate
10:25
Example Scatterplot
10:48
Shape, Trend, and Strength
10:49
Positive and Negative Association
14:05
Positive and Negative Association
14:06
Linearity, Strength, and Consistency
18:30
Linearity
18:31
Strength
19:14
Consistency
20:40
Summarizing a Scatterplot
22:58
Summarizing a Scatterplot
22:59
Example 1: Gapminder.org, Income x Life Expectancy
26:32
Example 2: Gapminder.org, Income x Infant Mortality
36:12
Example 3: Trend and Strength of Variables
40:14
Example 4: Trend, Strength and Shape for Scatterplots
43:27
Regression

32m 2s

Intro
0:00
0:05
0:06
Linear Equations
0:34
Linear Equations: y = mx + b
0:35
Rough Line
5:16
Rough Line
5:17
Regression - A 'Center' Line
7:41
Reasons for Summarizing with a Regression Line
7:42
Predictor and Response Variable
10:04
Goal of Regression
12:29
Goal of Regression
12:30
Prediction
14:50
Example: Servings of Mile Per Year Shown By Age
14:51
Intrapolation
17:06
Extrapolation
17:58
Error in Prediction
20:34
Prediction Error
20:35
Residual
21:40
Example 1: Residual
23:34
Example 2: Large and Negative Residual
26:30
Example 3: Positive Residual
28:13
Example 4: Interpret Regression Line & Extrapolate
29:40
Least Squares Regression

56m 36s

Intro
0:00
0:13
0:14
Best Fit
0:47
Best Fit
0:48
Sum of Squared Errors (SSE)
1:50
Sum of Squared Errors (SSE)
1:51
Why Squared?
3:38
Why Squared?
3:39
Quantitative Properties of Regression Line
4:51
Quantitative Properties of Regression Line
4:52
So How do we Find Such a Line?
6:49
SSEs of Different Line Equations & Lowest SSE
6:50
Carl Gauss' Method
8:01
How Do We Find Slope (b1)
11:00
How Do We Find Slope (b1)
11:01
Hoe Do We Find Intercept
15:11
Hoe Do We Find Intercept
15:12
Example 1: Which of These Equations Fit the Above Data Best?
17:18
Example 2: Find the Regression Line for These Data Points and Interpret It
26:31
Example 3: Summarize the Scatterplot and Find the Regression Line.
34:31
Example 4: Examine the Mean of Residuals
43:52
Correlation

43m 58s

Intro
0:00
0:05
0:06
Summarizing a Scatterplot Quantitatively
0:47
Shape
0:48
Trend
1:11
Strength: Correlation ®
1:45
Correlation Coefficient ( r )
2:30
Correlation Coefficient ( r )
2:31
Trees vs. Forest
11:59
Trees vs. Forest
12:00
Calculating r
15:07
Average Product of z-scores for x and y
15:08
Relationship between Correlation and Slope
21:10
Relationship between Correlation and Slope
21:11
Example 1: Find the Correlation between Grams of Fat and Cost
24:11
Example 2: Relationship between r and b1
30:24
Example 3: Find the Regression Line
33:35
Example 4: Find the Correlation Coefficient for this Set of Data
37:37
Correlation: r vs. r-squared

52m 52s

Intro
0:00
0:07
0:08
R-squared
0:44
What is the Meaning of It? Why Squared?
0:45
Parsing Sum of Squared (Parsing Variability)
2:25
SST = SSR + SSE
2:26
What is SST and SSE?
7:46
What is SST and SSE?
7:47
r-squared
18:33
Coefficient of Determination
18:34
If the Correlation is Strong…
20:25
If the Correlation is Strong…
20:26
If the Correlation is Weak…
22:36
If the Correlation is Weak…
22:37
Example 1: Find r-squared for this Set of Data
23:56
Example 2: What Does it Mean that the Simple Linear Regression is a 'Model' of Variance?
33:54
Example 3: Why Does r-squared Only Range from 0 to 1
37:29
Example 4: Find the r-squared for This Set of Data
39:55
Transformations of Data

27m 8s

Intro
0:00
0:05
0:06
Why Transform?
0:26
Why Transform?
0:27
Shape-preserving vs. Shape-changing Transformations
5:14
Shape-preserving = Linear Transformations
5:15
Shape-changing Transformations = Non-linear Transformations
6:20
Common Shape-Preserving Transformations
7:08
Common Shape-Preserving Transformations
7:09
Common Shape-Changing Transformations
8:59
Powers
9:00
Logarithms
9:39
Change Just One Variable? Both?
10:38
Log-log Transformations
10:39
Log Transformations
14:38
Example 1: Create, Graph, and Transform the Data Set
15:19
Example 2: Create, Graph, and Transform the Data Set
20:08
Example 3: What Kind of Model would You Choose for this Data?
22:44
Example 4: Transformation of Data
25:46
Section 6: Collecting Data in an Experiment
Sampling & Bias

54m 44s

Intro
0:00
0:05
0:06
Descriptive vs. Inferential Statistics
1:04
Descriptive Statistics: Data Exploration
1:05
Example
2:03
To tackle Generalization…
4:31
Generalization
4:32
Sampling
6:06
'Good' Sample
6:40
Defining Samples and Populations
8:55
Population
8:56
Sample
11:16
Why Use Sampling?
13:09
Why Use Sampling?
13:10
Goal of Sampling: Avoiding Bias
15:04
What is Bias?
15:05
Where does Bias Come from: Sampling Bias
17:53
Where does Bias Come from: Response Bias
18:27
Sampling Bias: Bias from Bas Sampling Methods
19:34
Size Bias
19:35
Voluntary Response Bias
21:13
Convenience Sample
22:22
Judgment Sample
23:58
25:40
Response Bias: Bias from 'Bad' Data Collection Methods
28:00
Nonresponse Bias
29:31
Questionnaire Bias
31:10
Incorrect Response or Measurement Bias
37:32
Example 1: What Kind of Biases?
40:29
Example 2: What Biases Might Arise?
44:46
Example 3: What Kind of Biases?
48:34
Example 4: What Kind of Biases?
51:43
Sampling Methods

14m 25s

Intro
0:00
0:05
0:06
Biased vs. Unbiased Sampling Methods
0:32
Biased Sampling
0:33
Unbiased Sampling
1:13
Probability Sampling Methods
2:31
Simple Random
2:54
Stratified Random Sampling
4:06
Cluster Sampling
5:24
Two-staged Sampling
6:22
Systematic Sampling
7:25
8:33
Example 2: Describe How to Take a Two-Stage Sample from this Book
10:16
Example 3: Sampling Methods
11:58
Example 4: Cluster Sample Plan
12:48
Research Design

53m 54s

Intro
0:00
0:06
0:07
Descriptive vs. Inferential Statistics
0:51
Descriptive Statistics: Data Exploration
0:52
Inferential Statistics
1:02
Variables and Relationships
1:44
Variables
1:45
Relationships
2:49
Not Every Type of Study is an Experiment…
4:16
Category I - Descriptive Study
4:54
Category II - Correlational Study
5:50
Category III - Experimental, Quasi-experimental, Non-experimental
6:33
Category III
7:42
Experimental, Quasi-experimental, and Non-experimental
7:43
Why CAN'T the Other Strategies Determine Causation?
10:18
Third-variable Problem
10:19
Directionality Problem
15:49
What Makes Experiments Special?
17:54
Manipulation
17:55
Control (and Comparison)
21:58
Methods of Control
26:38
Holding Constant
26:39
Matching
29:11
Random Assignment
31:48
Experiment Terminology
34:09
'true' Experiment vs. Study
34:10
Independent Variable (IV)
35:16
Dependent Variable (DV)
35:45
Factors
36:07
Treatment Conditions
36:23
Levels
37:43
Confounds or Extraneous Variables
38:04
Blind
38:38
Blind Experiments
38:39
Double-blind Experiments
39:29
How Categories Relate to Statistics
41:35
Category I - Descriptive Study
41:36
Category II - Correlational Study
42:05
Category III - Experimental, Quasi-experimental, Non-experimental
42:43
Example 1: Research Design
43:50
Example 2: Research Design
47:37
Example 3: Research Design
50:12
Example 4: Research Design
52:00
Between and Within Treatment Variability

41m 31s

Intro
0:00
0:06
0:07
Experimental Designs
0:51
Experimental Designs: Manipulation & Control
0:52
Two Types of Variability
2:09
Between Treatment Variability
2:10
Within Treatment Variability
3:31
Updated Goal of Experimental Design
5:47
Updated Goal of Experimental Design
5:48
Example: Drugs and Driving
6:56
Example: Drugs and Driving
6:57
Different Types of Random Assignment
11:27
All Experiments
11:28
Completely Random Design
12:02
Randomized Block Design
13:19
Randomized Block Design
15:48
Matched Pairs Design
15:49
Repeated Measures Design
19:47
Between-subject Variable vs. Within-subject Variable
22:43
Completely Randomized Design
22:44
Repeated Measures Design
25:03
Example 1: Design a Completely Random, Matched Pair, and Repeated Measures Experiment
26:16
Example 2: Block Design
31:41
Example 3: Completely Randomized Designs
35:11
Example 4: Completely Random, Matched Pairs, or Repeated Measures Experiments?
39:01
Section 7: Review of Probability Axioms
Sample Spaces

37m 52s

Intro
0:00
0:07
0:08
Why is Probability Involved in Statistics
0:48
Probability
0:49
Can People Tell the Difference between Cheap and Gourmet Coffee?
2:08
Taste Test with Coffee Drinkers
3:37
If No One can Actually Taste the Difference
3:38
If Everyone can Actually Taste the Difference
5:36
Creating a Probability Model
7:09
Creating a Probability Model
7:10
D'Alembert vs. Necker
9:41
D'Alembert vs. Necker
9:42
Problem with D'Alembert's Model
13:29
Problem with D'Alembert's Model
13:30
Covering Entire Sample Space
15:08
Fundamental Principle of Counting
15:09
Where Do Probabilities Come From?
22:54
Observed Data, Symmetry, and Subjective Estimates
22:55
Checking whether Model Matches Real World
24:27
Law of Large Numbers
24:28
Example 1: Law of Large Numbers
27:46
Example 2: Possible Outcomes
30:43
Example 3: Brands of Coffee and Taste
33:25
Example 4: How Many Different Treatments are there?
35:33

20m 29s

Intro
0:00
0:08
0:09
Disjoint Events
0:41
Disjoint Events
0:42
Meaning of 'or'
2:39
In Regular Life
2:40
In Math/Statistics/Computer Science
3:10
3:55
If A and B are Disjoint: P (A and B)
3:56
If A and B are Disjoint: P (A or B)
5:15
5:41
5:42
8:31
If A and B are not Disjoint: P (A or B)
8:32
Example 1: Which of These are Mutually Exclusive?
10:50
Example 2: What is the Probability that You will Have a Combination of One Heads and Two Tails?
12:57
Example 3: Engagement Party
15:17
Example 4: Home Owner's Insurance
18:30
Conditional Probability

57m 19s

Intro
0:00
0:05
0:06
'or' vs. 'and' vs. Conditional Probability
1:07
'or' vs. 'and' vs. Conditional Probability
1:08
'and' vs. Conditional Probability
5:57
P (M or L)
5:58
P (M and L)
8:41
P (M|L)
11:04
P (L|M)
12:24
Tree Diagram
15:02
Tree Diagram
15:03
Defining Conditional Probability
22:42
Defining Conditional Probability
22:43
Common Contexts for Conditional Probability
30:56
Medical Testing: Positive Predictive Value
30:57
Medical Testing: Sensitivity
33:03
Statistical Tests
34:27
Example 1: Drug and Disease
36:41
Example 2: Marbles and Conditional Probability
40:04
Example 3: Cards and Conditional Probability
45:59
Example 4: Votes and Conditional Probability
50:21
Independent Events

24m 27s

Intro
0:00
0:05
0:06
Independent Events & Conditional Probability
0:26
Non-independent Events
0:27
Independent Events
2:00
Non-independent and Independent Events
3:08
Non-independent and Independent Events
3:09
Defining Independent Events
5:52
Defining Independent Events
5:53
Multiplication Rule
7:29
Previously…
7:30
But with Independent Evens
8:53
Example 1: Which of These Pairs of Events are Independent?
11:12
Example 2: Health Insurance and Probability
15:12
Example 3: Independent Events
17:42
Example 4: Independent Events
20:03
Section 8: Probability Distributions
Introduction to Probability Distributions

56m 45s

Intro
0:00
0:08
0:09
Sampling vs. Probability
0:57
Sampling
0:58
Missing
1:30
What is Missing?
3:06
Insight: Probability Distributions
5:26
Insight: Probability Distributions
5:27
What is a Probability Distribution?
7:29
From Sample Spaces to Probability Distributions
8:44
Sample Space
8:45
Probability Distribution of the Sum of Two Die
11:16
The Random Variable
17:43
The Random Variable
17:44
Expected Value
21:52
Expected Value
21:53
Example 1: Probability Distributions
28:45
Example 2: Probability Distributions
35:30
Example 3: Probability Distributions
43:37
Example 4: Probability Distributions
47:20
Expected Value & Variance of Probability Distributions

53m 41s

Intro
0:00
0:06
0:07
Discrete vs. Continuous Random Variables
1:04
Discrete vs. Continuous Random Variables
1:05
Mean and Variance Review
4:44
Mean: Sample, Population, and Probability Distribution
4:45
Variance: Sample, Population, and Probability Distribution
9:12
Example Situation
14:10
Example Situation
14:11
Some Special Cases…
16:13
Some Special Cases…
16:14
Linear Transformations
19:22
Linear Transformations
19:23
What Happens to Mean and Variance of the Probability Distribution?
20:12
n Independent Values of X
25:38
n Independent Values of X
25:39
Compare These Two Situations
30:56
Compare These Two Situations
30:57
Two Random Variables, X and Y
32:02
Two Random Variables, X and Y
32:03
Example 1: Expected Value & Variance of Probability Distributions
35:35
Example 2: Expected Values & Standard Deviation
44:17
Example 3: Expected Winnings and Standard Deviation
48:18
Binomial Distribution

55m 15s

Intro
0:00
0:05
0:06
Discrete Probability Distributions
1:42
Discrete Probability Distributions
1:43
Binomial Distribution
2:36
Binomial Distribution
2:37
Multiplicative Rule Review
6:54
Multiplicative Rule Review
6:55
How Many Outcomes with k 'Successes'
10:23
Adults and Bachelor's Degree: Manual List of Outcomes
10:24
P (X=k)
19:37
Putting Together # of Outcomes with the Multiplicative Rule
19:38
Expected Value and Standard Deviation in a Binomial Distribution
25:22
Expected Value and Standard Deviation in a Binomial Distribution
25:23
Example 1: Coin Toss
33:42
38:03
Example 3: Types of Blood and Probability
45:39
Example 4: Expected Number and Standard Deviation
51:11
Section 9: Sampling Distributions of Statistics
Introduction to Sampling Distributions

48m 17s

Intro
0:00
0:08
0:09
Probability Distributions vs. Sampling Distributions
0:55
Probability Distributions vs. Sampling Distributions
0:56
Same Logic
3:55
Logic of Probability Distribution
3:56
Example: Rolling Two Die
6:56
Simulating Samples
9:53
To Come Up with Probability Distributions
9:54
In Sampling Distributions
11:12
Connecting Sampling and Research Methods with Sampling Distributions
12:11
Connecting Sampling and Research Methods with Sampling Distributions
12:12
Simulating a Sampling Distribution
14:14
Experimental Design: Regular Sleep vs. Less Sleep
14:15
Logic of Sampling Distributions
23:08
Logic of Sampling Distributions
23:09
General Method of Simulating Sampling Distributions
25:38
General Method of Simulating Sampling Distributions
25:39
Questions that Remain
28:45
Questions that Remain
28:46
Example 1: Mean and Standard Error of Sampling Distribution
30:57
Example 2: What is the Best Way to Describe Sampling Distributions?
37:12
Example 3: Matching Sampling Distributions
38:21
Example 4: Mean and Standard Error of Sampling Distribution
41:51
Sampling Distribution of the Mean

1h 8m 48s

Intro
0:00
0:05
0:06
Special Case of General Method for Simulating a Sampling Distribution
1:53
Special Case of General Method for Simulating a Sampling Distribution
1:54
Computer Simulation
3:43
Using Simulations to See Principles behind Shape of SDoM
15:50
Using Simulations to See Principles behind Shape of SDoM
15:51
Conditions
17:38
Using Simulations to See Principles behind Center (Mean) of SDoM
20:15
Using Simulations to See Principles behind Center (Mean) of SDoM
20:16
Conditions: Does n Matter?
21:31
Conditions: Does Number of Simulation Matter?
24:37
Using Simulations to See Principles behind Standard Deviation of SDoM
27:13
Using Simulations to See Principles behind Standard Deviation of SDoM
27:14
Conditions: Does n Matter?
34:45
Conditions: Does Number of Simulation Matter?
36:24
Central Limit Theorem
37:13
SHAPE
38:08
CENTER
39:34
39:52
Comparing Population, Sample, and SDoM
43:10
Comparing Population, Sample, and SDoM
43:11
48:24
What Happens When We Don't Know What the Population Looks Like?
48:25
Can We Have Sampling Distributions for Summary Statistics Other than the Mean?
49:42
How Do We Know whether a Sample is Sufficiently Unlikely?
53:36
Do We Always Have to Simulate a Large Number of Samples in Order to get a Sampling Distribution?
54:40
Example 1: Mean Batting Average
55:25
Example 2: Mean Sampling Distribution and Standard Error
59:07
Example 3: Sampling Distribution of the Mean
1:01:04
Sampling Distribution of Sample Proportions

54m 37s

Intro
0:00
0:06
0:07
Intro to Sampling Distribution of Sample Proportions (SDoSP)
0:51
Categorical Data (Examples)
0:52
Wish to Estimate Proportion of Population from Sample…
2:00
Notation
3:34
Population Proportion and Sample Proportion Notations
3:35
What's the Difference?
9:19
SDoM vs. SDoSP: Type of Data
9:20
SDoM vs. SDoSP: Shape
11:24
SDoM vs. SDoSP: Center
12:30
15:34
Binomial Distribution vs. Sampling Distribution of Sample Proportions
19:14
Binomial Distribution vs. SDoSP: Type of Data
19:17
Binomial Distribution vs. SDoSP: Shape
21:07
Binomial Distribution vs. SDoSP: Center
21:43
24:08
Example 1: Sampling Distribution of Sample Proportions
26:07
Example 2: Sampling Distribution of Sample Proportions
37:58
Example 3: Sampling Distribution of Sample Proportions
44:42
Example 4: Sampling Distribution of Sample Proportions
45:57
Section 10: Inferential Statistics
Introduction to Confidence Intervals

42m 53s

Intro
0:00
0:06
0:07
Inferential Statistics
0:50
Inferential Statistics
0:51
Two Problems with This Picture…
3:20
Two Problems with This Picture…
3:21
Solution: Confidence Intervals (CI)
4:59
Solution: Hypotheiss Testing (HT)
5:49
Which Parameters are Known?
6:45
Which Parameters are Known?
6:46
Confidence Interval - Goal
7:56
When We Don't Know m but know s
7:57
When We Don't Know
18:27
When We Don't Know m nor s
18:28
Example 1: Confidence Intervals
26:18
Example 2: Confidence Intervals
29:46
Example 3: Confidence Intervals
32:18
Example 4: Confidence Intervals
38:31
t Distributions

1h 2m 6s

Intro
0:00
0:04
0:05
When to Use z vs. t?
1:07
When to Use z vs. t?
1:08
What is z and t?
3:02
z-score and t-score: Commonality
3:03
z-score and t-score: Formulas
3:34
z-score and t-score: Difference
5:22
Why not z? (Why t?)
7:24
Why not z? (Why t?)
7:25
But Don't Worry!
15:13
Gossett and t-distributions
15:14
Rules of t Distributions
17:05
t-distributions are More Normal as n Gets Bigger
17:06
t-distributions are a Family of Distributions
18:55
Degrees of Freedom (df)
20:02
Degrees of Freedom (df)
20:03
t Family of Distributions
24:07
t Family of Distributions : df = 2 , 4, and 60
24:08
df = 60
29:16
df = 2
29:59
How to Find It?
31:01
'Student's t-distribution' or 't-distribution'
31:02
Excel Example
33:06
Example 1: Which Distribution Do You Use? Z or t?
45:26
47:41
Example 3: t Distributions
52:15
Example 4: t Distributions , confidence interval, and mean
55:59
Introduction to Hypothesis Testing

1h 6m 33s

Intro
0:00
0:06
0:07
Issues to Overcome in Inferential Statistics
1:35
Issues to Overcome in Inferential Statistics
1:36
What Happens When We Don't Know What the Population Looks Like?
2:57
How Do We Know whether a sample is Sufficiently Unlikely
3:43
Hypothesizing a Population
6:44
Hypothesizing a Population
6:45
Null Hypothesis
8:07
Alternative Hypothesis
8:56
Hypotheses
11:58
Hypotheses
11:59
Errors in Hypothesis Testing
14:22
Errors in Hypothesis Testing
14:23
Steps of Hypothesis Testing
21:15
Steps of Hypothesis Testing
21:16
Single Sample HT ( When Sigma Available)
26:08
26:09
Step1
27:08
Step 2
27:58
Step 3
28:17
Step 4
32:18
Single Sample HT (When Sigma Not Available)
36:33
36:34
Step1: Hypothesis Testing
36:58
Step 2: Significance Level
37:25
Step 3: Decision Stage
37:40
Step 4: Sample
41:36
Sigma and p-value
45:04
Sigma and p-value
45:05
On tailed vs. Two Tailed Hypotheses
45:51
Example 1: Hypothesis Testing
48:37
Example 2: Heights of Women in the US
57:43
Example 3: Select the Best Way to Complete This Sentence
1:03:23
Confidence Intervals for the Difference of Two Independent Means

55m 14s

Intro
0:00
0:14
0:15
One Mean vs. Two Means
1:17
One Mean vs. Two Means
1:18
Notation
2:41
A Sample! A Set!
2:42
Mean of X, Mean of Y, and Difference of Two Means
3:56
SE of X
4:34
SE of Y
6:28
Sampling Distribution of the Difference between Two Means (SDoD)
7:48
Sampling Distribution of the Difference between Two Means (SDoD)
7:49
Rules of the SDoD (similar to CLT!)
15:00
Mean for the SDoD Null Hypothesis
15:01
Standard Error
17:39
When can We Construct a CI for the Difference between Two Means?
21:28
Three Conditions
21:29
Finding CI
23:56
One Mean CI
23:57
Two Means CI
25:45
Finding t
29:16
Finding t
29:17
Interpreting CI
30:25
Interpreting CI
30:26
Better Estimate of s (s pool)
34:15
Better Estimate of s (s pool)
34:16
Example 1: Confidence Intervals
42:32
Example 2: SE of the Difference
52:36
Hypothesis Testing for the Difference of Two Independent Means

50m

Intro
0:00
0:06
0:07
The Goal of Hypothesis Testing
0:56
One Sample and Two Samples
0:57
Sampling Distribution of the Difference between Two Means (SDoD)
3:42
Sampling Distribution of the Difference between Two Means (SDoD)
3:43
Rules of the SDoD (Similar to CLT!)
6:46
Shape
6:47
Mean for the Null Hypothesis
7:26
Standard Error for Independent Samples (When Variance is Homogenous)
8:18
Standard Error for Independent Samples (When Variance is not Homogenous)
9:25
Same Conditions for HT as for CI
10:08
Three Conditions
10:09
Steps of Hypothesis Testing
11:04
Steps of Hypothesis Testing
11:05
Formulas that Go with Steps of Hypothesis Testing
13:21
Step 1
13:25
Step 2
14:18
Step 3
15:00
Step 4
16:57
Example 1: Hypothesis Testing for the Difference of Two Independent Means
18:47
Example 2: Hypothesis Testing for the Difference of Two Independent Means
33:55
Example 3: Hypothesis Testing for the Difference of Two Independent Means
44:22
Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means

1h 14m 11s

Intro
0:00
0:09
0:10
The Goal of Hypothesis Testing
1:27
One Sample and Two Samples
1:28
Independent Samples vs. Paired Samples
3:16
Independent Samples vs. Paired Samples
3:17
Which is Which?
5:20
Independent SAMPLES vs. Independent VARIABLES
7:43
independent SAMPLES vs. Independent VARIABLES
7:44
T-tests Always…
10:48
T-tests Always…
10:49
Notation for Paired Samples
12:59
Notation for Paired Samples
13:00
Steps of Hypothesis Testing for Paired Samples
16:13
Steps of Hypothesis Testing for Paired Samples
16:14
Rules of the SDoD (Adding on Paired Samples)
18:03
Shape
18:04
Mean for the Null Hypothesis
18:31
Standard Error for Independent Samples (When Variance is Homogenous)
19:25
Standard Error for Paired Samples
20:39
Formulas that go with Steps of Hypothesis Testing
22:59
Formulas that go with Steps of Hypothesis Testing
23:00
Confidence Intervals for Paired Samples
30:32
Confidence Intervals for Paired Samples
30:33
Example 1: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
32:28
Example 2: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
44:02
Example 3: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
52:23
Type I and Type II Errors

31m 27s

Intro
0:00
0:18
0:19
Errors and Relationship to HT and the Sample Statistic?
1:11
Errors and Relationship to HT and the Sample Statistic?
1:12
7:00
One Sample t-test: Friends on Facebook
7:01
Two Sample t-test: Friends on Facebook
13:46
Usually, Lots of Overlap between Null and Alternative Distributions
16:59
Overlap between Null and Alternative Distributions
17:00
How Distributions and 'Box' Fit Together
22:45
How Distributions and 'Box' Fit Together
22:46
Example 1: Types of Errors
25:54
Example 2: Types of Errors
27:30
Example 3: What is the Danger of the Type I Error?
29:38
Effect Size & Power

44m 41s

Intro
0:00
0:05
0:06
Distance between Distributions: Sample t
0:49
Distance between Distributions: Sample t
0:50
Problem with Distance in Terms of Standard Error
2:56
Problem with Distance in Terms of Standard Error
2:57
Test Statistic (t) vs. Effect Size (d or g)
4:38
Test Statistic (t) vs. Effect Size (d or g)
4:39
Rules of Effect Size
6:09
Rules of Effect Size
6:10
Why Do We Need Effect Size?
8:21
Tells You the Practical Significance
8:22
HT can be Deceiving…
10:25
Important Note
10:42
What is Power?
11:20
What is Power?
11:21
Why Do We Need Power?
14:19
Conditional Probability and Power
14:20
Power is:
16:27
Can We Calculate Power?
19:00
Can We Calculate Power?
19:01
How Does Alpha Affect Power?
20:36
How Does Alpha Affect Power?
20:37
How Does Effect Size Affect Power?
25:38
How Does Effect Size Affect Power?
25:39
How Does Variability and Sample Size Affect Power?
27:56
How Does Variability and Sample Size Affect Power?
27:57
How Do We Increase Power?
32:47
Increasing Power
32:48
Example 1: Effect Size & Power
35:40
Example 2: Effect Size & Power
37:38
Example 3: Effect Size & Power
40:55
Section 11: Analysis of Variance
F-distributions

24m 46s

Intro
0:00
0:04
0:05
Z- & T-statistic and Their Distribution
0:34
Z- & T-statistic and Their Distribution
0:35
F-statistic
4:55
The F Ration ( the Variance Ratio)
4:56
F-distribution
12:29
F-distribution
12:30
s and p-value
15:00
s and p-value
15:01
Example 1: Why Does F-distribution Stop At 0 But Go On Until Infinity?
18:33
Example 2: F-distributions
19:29
Example 3: F-distributions and Heights
21:29
ANOVA with Independent Samples

1h 9m 25s

Intro
0:00
0:05
0:06
The Limitations of t-tests
1:12
The Limitations of t-tests
1:13
Two Major Limitations of Many t-tests
3:26
Two Major Limitations of Many t-tests
3:27
Ronald Fisher's Solution… F-test! New Null Hypothesis
4:43
Ronald Fisher's Solution… F-test! New Null Hypothesis (Omnibus Test - One Test to Rule Them All!)
4:44
Analysis of Variance (ANoVA) Notation
7:47
Analysis of Variance (ANoVA) Notation
7:48
Partitioning (Analyzing) Variance
9:58
Total Variance
9:59
Within-group Variation
14:00
Between-group Variation
16:22
Time out: Review Variance & SS
17:05
Time out: Review Variance & SS
17:06
F-statistic
19:22
The F Ratio (the Variance Ratio)
19:23
S²bet = SSbet / dfbet
22:13
What is This?
22:14
How Many Means?
23:20
So What is the dfbet?
23:38
So What is SSbet?
24:15
S²w = SSw / dfw
26:05
What is This?
26:06
How Many Means?
27:20
So What is the dfw?
27:36
So What is SSw?
28:18
Chart of Independent Samples ANOVA
29:25
Chart of Independent Samples ANOVA
29:26
Example 1: Who Uploads More Photos: Unknown Ethnicity, Latino, Asian, Black, or White Facebook Users?
35:52
Hypotheses
35:53
Significance Level
39:40
Decision Stage
40:05
Calculate Samples' Statistic and p-Value
44:10
Reject or Fail to Reject H0
55:54
Example 2: ANOVA with Independent Samples
58:21
Repeated Measures ANOVA

1h 15m 13s

Intro
0:00
0:05
0:06
The Limitations of t-tests
0:36
Who Uploads more Pictures and Which Photo-Type is Most Frequently Used on Facebook?
0:37
ANOVA (F-test) to the Rescue!
5:49
Omnibus Hypothesis
5:50
Analyze Variance
7:27
Independent Samples vs. Repeated Measures
9:12
Same Start
9:13
Independent Samples ANOVA
10:43
Repeated Measures ANOVA
12:00
Independent Samples ANOVA
16:00
Same Start: All the Variance Around Grand Mean
16:01
Independent Samples
16:23
Repeated Measures ANOVA
18:18
Same Start: All the Variance Around Grand Mean
18:19
Repeated Measures
18:33
Repeated Measures F-statistic
21:22
The F Ratio (The Variance Ratio)
21:23
S²bet = SSbet / dfbet
23:07
What is This?
23:08
How Many Means?
23:39
So What is the dfbet?
23:54
So What is SSbet?
24:32
S² resid = SS resid / df resid
25:46
What is This?
25:47
So What is SS resid?
26:44
So What is the df resid?
27:36
SS subj and df subj
28:11
What is This?
28:12
How Many Subject Means?
29:43
So What is df subj?
30:01
So What is SS subj?
30:09
SS total and df total
31:42
What is This?
31:43
What is the Total Number of Data Points?
32:02
So What is df total?
32:34
so What is SS total?
32:47
Chart of Repeated Measures ANOVA
33:19
Chart of Repeated Measures ANOVA: F and Between-samples Variability
33:20
Chart of Repeated Measures ANOVA: Total Variability, Within-subject (case) Variability, Residual Variability
35:50
Example 1: Which is More Prevalent on Facebook: Tagged, Uploaded, Mobile, or Profile Photos?
40:25
Hypotheses
40:26
Significance Level
41:46
Decision Stage
42:09
Calculate Samples' Statistic and p-Value
46:18
Reject or Fail to Reject H0
57:55
Example 2: Repeated Measures ANOVA
58:57
Example 3: What's the Problem with a Bunch of Tiny t-tests?
1:13:59
Section 12: Chi-square Test
Chi-Square Goodness-of-Fit Test

58m 23s

Intro
0:00
0:05
0:06
Where Does the Chi-Square Test Belong?
0:50
Where Does the Chi-Square Test Belong?
0:51
A New Twist on HT: Goodness-of-Fit
7:23
HT in General
7:24
Goodness-of-Fit HT
8:26
12:17
Null Hypothesis
12:18
Alternative Hypothesis
13:23
Example
14:38
Chi-Square Statistic
17:52
Chi-Square Statistic
17:53
Chi-Square Distributions
24:31
Chi-Square Distributions
24:32
Conditions for Chi-Square
28:58
Condition 1
28:59
Condition 2
30:20
Condition 3
30:32
Condition 4
31:47
Example 1: Chi-Square Goodness-of-Fit Test
32:23
Example 2: Chi-Square Goodness-of-Fit Test
44:34
Example 3: Which of These Statements Describe Properties of the Chi-Square Goodness-of-Fit Test?
56:06
Chi-Square Test of Homogeneity

51m 36s

Intro
0:00
0:09
0:10
Goodness-of-Fit vs. Homogeneity
1:13
Goodness-of-Fit HT
1:14
Homogeneity
2:00
Analogy
2:38
5:00
Null Hypothesis
5:01
Alternative Hypothesis
6:11
Example
6:33
Chi-Square Statistic
10:12
Same as Goodness-of-Fit Test
10:13
Set Up Data
12:28
Setting Up Data Example
12:29
Expected Frequency
16:53
Expected Frequency
16:54
Chi-Square Distributions & df
19:26
Chi-Square Distributions & df
19:27
Conditions for Test of Homogeneity
20:54
Condition 1
20:55
Condition 2
21:39
Condition 3
22:05
Condition 4
22:23
Example 1: Chi-Square Test of Homogeneity
22:52
Example 2: Chi-Square Test of Homogeneity
32:10
Section 13: Overview of Statistics
Overview of Statistics

18m 11s

Intro
0:00
0:07
0:08
The Statistical Tests (HT) We've Covered
0:28
The Statistical Tests (HT) We've Covered
0:29
Organizing the Tests We've Covered…
1:08
One Sample: Continuous DV and Categorical DV
1:09
Two Samples: Continuous DV and Categorical DV
5:41
More Than Two Samples: Continuous DV and Categorical DV
8:21
The Following Data: OK Cupid
10:10
The Following Data: OK Cupid
10:11
Example 1: Weird-MySpace-Angle Profile Photo
10:38
Example 2: Geniuses
12:30
Example 3: Promiscuous iPhone Users
13:37
Example 4: Women, Aging, and Messaging
16:07

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### Chi-Square Test of Homogeneity

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Goodness-of-Fit vs. Homogeneity 1:13
• Goodness-of-Fit HT
• Homogeneity
• Analogy
• Null Hypothesis
• Alternative Hypothesis
• Example
• Chi-Square Statistic 10:12
• Same as Goodness-of-Fit Test
• Set Up Data 12:28
• Setting Up Data Example
• Expected Frequency 16:53
• Expected Frequency
• Chi-Square Distributions & df 19:26
• Chi-Square Distributions & df
• Conditions for Test of Homogeneity 20:54
• Condition 1
• Condition 2
• Condition 3
• Condition 4
• Example 1: Chi-Square Test of Homogeneity 22:52
• Example 2: Chi-Square Test of Homogeneity 32:10

### Transcription: Chi-Square Test of Homogeneity

Hi, welcome to educator.com.0002

We are going to talk about the chi-square test of homogeneity.0002

Previously we talked about the chi-square goodness of fit test now were in a contrast that with this new test is still 0018.3 chi-square test but it is a test of homogeneity now.0005

We are going to try and figure out when do we use which test.0022

The test we are testing a new idea , we are not testing goodness of that would actually testing homogeneity similar.0027

We actually have slightly different null hypotheses and alternative null and alternative hypotheses .0035

We are going to talk about how those have changed then we are going to go over the chi-square statistic and also finding 0051.0 the expected values is going to be a little bit different in test of homogeneity .0041

Finally working to go through chi-square distributions as well as degrees of freedom and the conditions for the test of homogeneity,0055

one can you actually care conduct this test service statistically legally.0065

Okay so the first thing is what is the difference between the test of homogeneity and test of goodness of fit?0069

Well in the goodness of fit hypothesis testing we wanted to determine whether sample proportions are very different from hypothesized0082

population proportion one way you could think about this is that you have one sample and you are comparing it to some hypothetical population.0089

In test of homogeneity and I called it goodness of fit, it is about how well these two things fit together.0098

How well does the sample fit with the hypothesized proportion.0108

In test of homogeneity homogeneous means similar right, that they are made up of the same stuff.0112

In test of homogeneity we want to determine whether 2 populations that are sorted into categories share the same proportions or not.0120

And here you could also substitute this word population here because ultimately were using the sample as a proxy for the population.0130

So here we have 2 population and we want to know whether those two populations are similar in their proportions or not0142

right were not comparing them to some hypothesized population were comparing them to each other.0152

And so really you can think of this as an analogy you think of the their relationship by using an analogy from the0159

one sample to the independent samples t-test.0167

In the one sample t-test we had one sample and we compared it to the null hypothesis right?0170

That was when we would have null hypotheses such as new equals zero or new equals 200 or new equals -5 versus an independent sample.0176

We had 2 samples and we wanted to know how similar they were to each other right or how different0190

they were from each other and our null hypothesis was changed to something like use of X bar minus Y bar equals zero right,0198

that they are either made up of the same mean or different means.0208

And in a in a similar way the goodness of fit chi-square is really asking whether this proportion in my sample0213

is similar to the proportion in our population.0229

So that is how I am comparing , this is my null hypothesis in some ways .0232

In our inner test of homogeneity we have 2 sample 2 population 2 sample that come from 2 unknown population and we want to know0240

whether these have similar proportions to each other and so that is going to be our null hypothesis that these have the same proportion or have different one.0255

For null hypotheses is similar proportion.0267

And so in that way I hope you could see that goodness of fit in homogeneity their ideas that we have looked at before0275

comparing one sample to a hypothesized population or comparing two samples to each other but we have looked at it before0285

not with proportion but with means, right?0294

And now are looking at it with proportion okay since you are looking at proportion we should have hypotheses about0297

proportion so the null hypotheses with something like this the proportion of all the each category the proportion that0305

all into each category is the same for each population so however many categories you have so let us say we have0313

in a three categories.0322

If we believe that they are the same and they should roughly have the same proportion so these have similar proportion.0341

It does not actually matter what the proportions are it could be 90, 10 could be 10,10 it could be 75 20 like when the proportions0347

that were think there similar for each population and whatever 780 whatever category is 75% of the population0360

that category will also be 75% of the population.0368

The alternative hypothesis says that for at least one category the populations do not have the same proportion so just like before0371

were now talking about differences that the differences are really in the proportions the predicted the populations proportion.0383

So just to give you an example.0394

Here is the problem and let us try to change it into the null hypothesis as well as alternative hypothesis.0396

So according to a poll for and six Democrats said they were very satisfied with candidate A while 510 were unsatisfied0401

however 910 Republicans were satisfied with candidate a while 60 were not.0410

And in a chi-square test of homogeneity we could see whether the proportions of Democrats and Republicans that Democrats were satisfied are0415

similar to the proportions were Republican of Republicans were satisfied versus unsatisfied.0427

So let us draw this out first.0436

So here we have about 400 Democrats saying there satisfied while 500 saying unsatisfied.0439

Let put satisfied in blue and so that is a little bit less than half and the unsatisfied people are a little bit0451

more than half so this is the Democratic population that they look like.0460

The Republican population looks very different so here we see most of the Republicans being pretty satisfied and0467

only a very small minority being unsatisfied right.0479

And so the question is are these two are the two similar are the proportions that fall into each category0483

satisfied or unsatisfied the same for each population?0493

Are they different?0497

The null hypothesis would probably say something like this.0498

The proportion of satisfied and unsatisfied people like us are similar are the same for Dans as well as republicans.0501

The alternative hypothesis says for at least one category either satisfied or unsatisfied, Dans and Republicans do not have the same proportion.0531

Okay so note that in the case of 2, once category changes once the proportion of one category changes the other one automatically changes.0561

So if we somehow were able to change has satisfied the Democrats were with candidate A, we would also see the0584

proportion of unsatisfied people just automatically change.0592

So that is in the case of two categories but in the case of multiple categories maybe 2 might change but the others may0595

not change right so in that way this would be a more general way of saying alternative hypothesis.0606

Now let us talk about the chi-square statistic.0612

Now the nice thing about the chi-square statistic is that it is the same as the goodness of fit test.0616

We use the same idea so chi-square is going to be observed frequencies and the difference between that and0621

expected frequencies where over the proportion of expected frequency.0631

But there is just one subtle difference before it was for each category.0638

Now we have different categories in different population right so we not only have like category 1 and category 20643

category 3 so on and so forth but we also have population 1 and population 2 at least right?0651

And so we have multiple of observed frequencies and so what do we do right?0659

Well what we do here is that we consider each of these combination of which population your in and which category0668

are talking about each of these are going to be called cells.0681

And so we do this for each cell so I will go from one of to the number of cells.0686

And how do we get the number of cells?0694

Well the number of cells is really how many population right and that is usually shown in columns times how many categories.0701

And that is usually shown in rows, you can also think of the number of cells as columns times rows, how many columns you have times the number of rows.0718

But really the idea comes from how many different populations your comparing of chi-square test of homogeneity0733

actually compare three or four population not just 2 and how many categories you are comparing.0739

So in order to use the chi-square formula, it is often helpful to set up your data in a particular way often0747

though that often these formulas will refer to rows and columns and so you really need to have the right data in0758

the rows and the right data columns in order for any of these formulas to be used correctly.0764

So how to set up your data in this way?0769

Whatever your sample one is you want to put that all of the information for sample one into a column, right so0772

here I put sample 1 at the generic sample one it could be college freshmen are Democrats or mice got a certain0780

drive whatever it is the sample one and these are the people in sample 1 who fell into category one.0788

These are the people in sample 1 who fell in to category two and these are called cells.0798

When you add these frequency that you should get the total number of people in sample 1 right so in that way all0804

the information from 1 one is in a column.0814

Same thing with sample 2 all the information from sample 2 should be in a column.0818

This should be the entire sample broken up into those that fell into category 1 versus category two and then the0823

total gives you the total number of cases in sample 2.0830

If you had sample three and four they would follow that same pattern and all the information should be in one column.0836

On the flip side when you look at rows you should be able to count of how many people how many cases were in category one.0843

And so if you count them up this way this is a sample but it is just how many cases in the entire data set that you are looking at0855

are in category 1 and if you look across here this is how many cases in the entire data set fall into category 20868

and finally if you look at this total of totals what you should get is that is the entire data set all added up.0878

So let us try that here with the Democrats and Republican example.0889

So I am going to put Democrats appear Republicans appear satisfied and unsatisfied and all I need to do is make0896

sure I find the correct information and put it into the correct cells.0910

910 are satisfied 60 are not.0916

When I add this up I should be able to get the number of how many Democrats total that are in the sample so this0921

is 916 for Republicans this is 970 so we have slightly more people in a Republican sample than our Democrat sample and that is fine.0929

If I add the rows up like this if I get the row totals what I should get is just a number of satisfied people.0940

It does not matter whether their Democrats or Republicans so we should get 13, 16 and this should be 570.0948

in a different order so that should be 1886.0967

So we have 1886 in our total data set across both sample and we know how many people were satisfied , how many0973

people are unsatisfied we also know how many Democrats we had how many Republicans we have and all the different combination right?0990

Democrats are satisfied Democrats unsatisfied Republican satisfied Republicans unsatisfied.0998

So this is a great way to set up your data that really can help you figure out expected frequency which is a1003

little bit more complicated to figure out intensive homogeneity.1009

Not too much complicated but just a little bit more.1012

So here is how we can figure out expected frequency so once you have it set up in this way Democrats Republicans1017

satisfied unsatisfied, once you have it set up in this way here is the formula used for expected frequency.1026

So E is going to equal basically the proportion of people who are in one particular category.1033

So I just want to know how people tend to be satisfied.1042

I do not care whether their across a Republican, just in general who satisfied right so that would be the row1046

total right so the row total over the grand total this one right here.1053

This will give me the rates or the proportion of just the general rate of who satisfied who tends to be satisfied1065

that 70% to be satisfied 20% to be satisfied 95% to be satisfied.1077

What is the general rate and I am going to multiply that by the total number of the sample that I am interested in1084

so maybe I am interested in the Democratic sample so I would get the column totals.1092

So that is the general formula that will show you this in a more specific way so let us talk about the expected value of1097

Democrats who are satisfied.1107

Right so that would be the satisfied total over the grand total so this gives us the rates of being satisfied just1110

in general what proportion of the entire data set is satisfied and then I am going to multiply that by however1125

many Democrats I have so Democrat total.1132

So I could write it in this way but what ends up is that this is just a more general way of saying this example.1137

So when I say Democrats total is the same thing as being column totals.1146

And when I say row total it is really the same thing as being satisfied total and the grand total is the total number in our data set.1151

Democrats Republicans.1162

So now let us talk about once you have the expected values you have the observed frequencies and now you could easily find chi-square.1165

Once you get your chi-square how do you compare it to the chi-square distribution?1176

Well the nice thing is the chi-square distribution looks the same as in the test at as in the goodness of fit test1182

and so chi-square it has a wall at zero can not be lower than zero and it has a long positive tail and when you decide how much1190

your alpha is and that is what it is going to look like Alpha is always one tailed in a chi-square distribution.1202

But the question is how to find degrees of freedom now that we have rows and columns?1208

Well the degrees of freedom is really going to be the degrees of freedom for category times the degrees of freedom for1217

however many populations or sample that represent your population you have and that is going to be the number of rows1229

right because each categories in a row -1 times the number of columns you have -1 so that is how you find you degrees of freedom1238

when you have more than one population that you are comparing.1248

So what are the conditions for the test of homogeneity?1251

These conditions are to be very similar to the conditions for out goodness of fit testing so the first thing is1258

each outcome of each population falls into exactly one of the fixed number of category.1265

Well the categories are mutually exclusive just like before, you have to be in one or the other you cannot be into 2 categories1275

at the same time you cannot opt out of being in a category also the category choices must be the same for all population.1280

So it went to one population has to have if they have three choices the same three choices must be the case for population 2.1288

The 2nd requirement for condition is that you must have independent and random sample before in tests of goodness of fit1298

we only have this requirement that the sample have to be branded because we only had one sample.1310

Now we have multiple samples and they must be independent of each other they cannot they cannot come from the same pool.1315

So third condition the expected frequency in each cell is five or greater and not just is the same condition that we had1325

for goodness of fit testing it is because you want a big a sample as well as the big enough proportion.1337

And number four is not really a condition is just so that you know how free you are with chi-square testing you can have1344

more than two categories and you can have more than two populations you could have 4 categories and six population so you1355

should have a whole bunch of these different combination so you are not restricted to 2 categories and 2 population.1364

So now let us go on to some examples.1371

Example 1 is just the example we have been using to talk about how to find how to set up your data and how to find1376

expected values so I set this up in an Excel file this is just exactly the same way we set it up previously I just found1383

the row totals as well as the column totals.1397

And now I could start of my hypothesis testing so first things first.1400

Step one our null hypothesis should say something like this that the proportions of satisfied and unsatisfied people minus adults1406

for Democrats should be the same as for Republican so the proportion of category one and two of satisfied and1425

unsatisfied by Allstate voters should be similar for Democrat and Republican.1435

So the alternative hypothesis is that at least one of those proportion will be different between Democrats and Republicans.1446

Step two, just set our alpha to be .05 and we know that because we are doing chi-square hypothesis testing is one1461

tailed step three you might want to draw a chi-square distribution for yourself or just in your head and certain1476

color and that alpha part and try to think.1485

I want to find my critical chi-square.1488

In order to find the critical chi-square I need to find the degrees of freedom.1493

And my degrees of freedom is going to be made up of the degrees of freedom for categories as well as the degree of1499

nfreedom for population and there is two populations so it is 2-1 and you could also see that as the columns 2 column – 1.1509

And the degrees of freedom for number of categories is with two categories that is satisfied and unsatisfied -11521

and there that corresponds perfectly to number of rows -1 and so the degrees of freedom here is going to be that1535

this times this so degrees of freedom for category times degrees of freedom for population and is just one.1545

So, what is our critical chi-square, but that is going to be found by chi in we put in our probability as well as1553

our degrees of freedom and we find 3.84 is our chi-square critical chi-square.1564

So we are looking for sample that represent population sample chi-square is that are larger than 3.84.1571

Step four look something like this so in order to find your sample chi-square what we need to do first is find our1584

expected values so here we have observed frequency and what we need to do is find infected frequency.1595

So I am just going to copy and paste this down here so we do not have to keep scrolling and so I am going to draw1609

a director at the table here for observed frequency and create the same table for expected frequency.1623

Okay so when I look at my expected frequency I need to find out what is the general rate and then multiply it by1635

however many however many industry people have in that sample so the general rate of being satisfied is 1316÷18861651

so that the general rate and that is about 70%.1670

Take that and multiply that by the total number of Democrats.1674

Now this part I want to keep that the same and I want to keep that in the same column so I am going to put \$ affinity1680

to walk down that column and here I am going to put \$ in front of both the D and the 21 in order to lock down this actual cell.1697

Because here is what I am going to do I am than actually copy and paste that over here and if look at this then what I am doing1708

is I have this same rates again the rate of being satisfied but now it is multiplied by the number of total Republicans.1716

And I am going to take that cell copy and paste it down here and here I see that now I have the rates of being1726

unsatisfied and they need to change this to that and here I have the rates of being unsatisfied and then1737

multiplied by total number of Republican so these are my expected frequencies.1750

Notice that the total still add up to be the same right and usually it should there might be some slight discrepancies1756

but that will just be because of rounding error so they should still be pretty close.1766

So now we have observed frequencies as well as expected frequencies and now we need to figure out my chi-square.1771

My chi-square is going to be made up of observed frequency minus expected frequency squared divided by expected frequency.1779

And I am going to need to find that for Democrat Republican as well as satisfied and unsatisfied and then add off all of these cells.1790

So I will see grand total and I will put that over here.1808

Okay so let us find the observed frequency minus the expected frequency squared divided by expected frequency.1813

And I could just copy and paste that here because Excel will just move everything down and I can take this over here because Excel1829

will move everything over to the right.1841

And the grand total for all four of these is going to be 547.18 and so my sample chi-square is quite large.1843

And so do I reject my no hypothesis?1876

Indeed I do and we can find the P value so here I will put chi disc in order to find my probability.1881

Here it is, degrees of freedom is going to be one and that is a very very very small P value so that is the pretty radically1898

different population that we set in there.1911

And if you want to step five, example 2.1917

Consider this data on pesticide residue on domestic and imported fruits.1933

Does this data fit the conditions of a chi-square test of homogeneity regardless of your answer conduct hypothesis tests.1937

Now be careful here although you see column and rows these are not the columns and rows you should be using the columns are1944

actually okay domestic roads imported roads we could consider those two to be the different populations that are interested in.1956

But the roads actually do not show the different categories such as sample size percentage showing no residue and percentage showing residue in violation right?1964

So what we should do is we should actually transform this data into sort of the correct setup.1975

So here you could just pull up a brand-new XL file just been a user of the bottom portion here and here is what we want.1983

We would like it to be set up so that we have the two populations appear and we have the different categories here2005

so the categories are probably going to be showing no residue showing residue in violation but one of the things I2028

noticed is that these percentages do not add up to 100 that there must be some other category that were missing.2035

So no residue showing residue in violation of the law so I guess that is really bad and maybe there is just one2042

word it is residue but not in violation and you sort of have to figure that out from the data that they have given you.2054

But they do give you the sample size 344 as well as 1136 so this is the total.2063

The question is what are our observed value?2073

In order to find observed value all we have to do is multiply but the proportion so 44.2% times the total.2079

Here I walk down that row, now residue in violation what I have to do is to change this percentage so the percentage is .9%.2098

So that is .009 so that is .9%.2116

And so what sort of leftover?2127

Well, the leftover percentages is 1-.442 + .009 right so that sort of everybody else and that is I guess the2131

number of fruits that are not in violation but still have some residue on them, some pesticide residue times this.2143

And so when I add them all up I could check and that is 344 so I have done my proportions correctly.2154

Now right away we could see that were actually not meeting the conditions for chi-square.2169

If you look at this cell right here that has that only has three fruits in it even if we round up generously it is 3.1 right?2176

So there is only three fruits.2188

Remember expected frequencies have to have at least 5, so here the observed value is pretty small.2191

Okay so that it said go ahead into hypothesis testing anyway you should not do this in real life but2200

for the purpose of this exercise let us do it.2210

So now let us find the proportion of imported fruits that are observed to have no residue on them.2212

So that 70% 70.4% times this total and that is almost 800 fruits.2222

Also we have those that have residue in violation .036 that is 3.6% times 1136, about 41 fruits and then2232

I need the leftover percentage , so that is 1-.70% 74.4% +3.6% .2249

That percentage times the total.2262

And that is 295 right?2268

So first notice that these seem like there is way more of these imported fruit than domestic fruits but that is because the2272

totals are different so it does not necessarily mean that imported fruits they have so much residue on them,2280

that is not necessarily what it means, but that is hard to compare because they have totally different totals.2289

So it is helpful to find the row totals as well because that can help us find expected value expected frequency2299

and so that is adding these rows together and we have a total of 1480 fruits Domestic and imported altogether.2308

Once we have that then it would be easy for us to find expected frequency and expected frequency we could basically set up in a very similar way.2329

So what is our expected frequency?2346

Well,expected frequency is generally how frequent with the proportion of no residue over all the fruits right.2362

So that will be this row totals divided by the grand total that is the general rates and we want to lockdown this row2370

because we want to lock those two values down because and that is always going to be the rate for no residue2383

times the actual number of domestic fruits.2401

So we get 221 and here we do the same thing and I just copied and pasted across an Excel will just naturally you figure out what to do.2410

So this is the rate of no residue over total fruits times the total number of imported fruits.2428

Then we find there the rates of fruits that have residue but are not in violation which is this total over the grand total.2436

And then I am going to lockdown those values and then I am going to multiply that by the total number of domestic fruit.2449

And then if I copy that over that should give me the total number of imported fruits expected value of imported fruits given this proportion.2467

And finally the proportion of fruits with residue in violation so a lot of pesticide residue that would be this total2476

divided by the grand total times the total.2489

And here what we can see is if we sum these three expected frequency together we should get something similar to 344.2502

And indeed we do and here we should be 1136 and indeed we do great.2515

So once we have our table of observed frequencies as well as expected frequencies now we can start to calculate2522

for each cell the observed frequency minus expected frequencies where as a proportion of expected frequency.2530

So O minus E squared as a proportion of expected frequency so I will copy this cell labels so observed frequency2540

minus expected frequency squared divided by expected frequency , and just copy and paste all that let us check one of this.2558

This one says that observed frequency minus expected frequency squared over expected frequency.2573

And when we add all of these up we get 102 but we have forgotten the difference as we forgot to make a decision stage2581

so let us go ahead and do step three.2599

So the decision stage will be our critical chi-square and our critical chi-square sound with degrees of freedom2601

of the categories times the degrees of freedom of the population multiplied together so the other degrees of freedom for the chi-square.2610

So categories -1 is 2, population -1 is 1, so the degrees of freedom is just 2, so our critical chi-square is chi in.2628

Put in .05 as our desired probability, our degrees of freedom equals 2 and we get 5.99.2646

We see that our chi-square is much larger than that so we would reject our null.2653

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